This paper presents mathematical properties of solutions to the Navier-Stokes equations for compressible fluids. We first review existence results for the Cauchy problem, and describe some regularity properties of solutions in the presence of possibly vanishing densities. Finally, we address the problem of the low Mach number limit leading to incompressible models.

We introduce a notion of topological synchronization by using Poincare recurrences of coupled subsystems. We show that the dimension of Poincare recurrences may indicate synchronized behavior.

A slant immersion was introduced in [1] as an isometric immersion of a Riemannian manifold into an almost Hermitian manifold with constant Wirtinger angle. It is known that there exist ample examples of slant submanifolds; in particular, slant surfaces in complex-spaceforms. In this paper, we establish a sharp inequality for slant surfaces and determine the Riemannian structures of special slant surfaces in complexspace- forms. By applying the special forms of the Riemannian structures on special slant surfaces we prove that proper slant surfaces in C2 are minimal if and only if they are special slant. We also determine proper slant surfaces in complex-space-forms which satisfy the equality case of the inequality identically.

Of concern are singular ordinary dierential operators de- ned on an interval for which the coecient of ellipticity degenerates at the boundary of the interval. Thus, for instance, of concern is the operator A dened by Au := u00 + u0 on the interval (0; 1) where 2 C[0; 1]; $\gt $ 0 in (0; 1) and (0) = 0 = (1): We show that, in many cases, A (with suitable boundary conditions) generates a C0-semigroup on various (weighted or not) Lp spaces on (0; 1) which is analytic. It is the analyticity that is the focal point.

In this work, we give conditions for the superposition of independent point processes to converge to a nonhomogeneous twodimensional Poisson process. Using our results, not only that of Grigelionis (1963) and Weissman (1975), but also some limiting results related to thinned point processes can be obtained. Hence this establishes the connection between thinnings and superpositions of point processes.

The present paper discusses the density of quotients from two dierent Muntz systems. An interesting and nontrivial generalization of a result of Somorjai is established.

This note presents general formulae for the lower bound of the rst two Dirichlet eigenvalues on a regular domain. As applications, the positivity of top spectrum and the gap between these two Dirichlet eigenvalues are studied.

In this paper, we investigate weakly compact and compact derivations of certain CSL algebras. These algebras contain nite tensor product algebras of nest algebras and some nest subalgebras of a von Neumann algebra.